Summary:In this paper rings for which every $s$-torsion quasi-injective module is weakly $s$-divisible for a hereditary preradical $s$ are characterized in terms of the properties of the corresponding lattice of the (hereditary) preradicals. In case of a stable torsion theory these rings coincide with $TQI$-rings investigated by J. Ahsan and E. Enochs in [1]. Our aim was to generalize some results concerning $QI$-rings obtained by J.S. Golan and S.R. L'opez-Permouth in [12]. A characterization of the $QI$-property in the category $\sigma[M]$ then follows as a consequence.